80.5.4 problem 5

Internal problem ID [18483]
Book : Elementary Differential Equations. By Thornton C. Fry. D Van Nostrand. NY. First Edition (1929)
Section : Chapter IV. Methods of solution: First order equations. section 32. Problems at page 89
Problem number : 5
Date solved : Thursday, March 13, 2025 at 12:06:24 PM
CAS classification : [[_homogeneous, `class G`]]

\begin{align*} \sqrt {t^{2}+T}&=T^{\prime } \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 136

ode:=(t^2+T(t))^(1/2) = diff(T(t),t); 
dsolve(ode,T(t), singsol=all);
 

\[ 17 \ln \left (-t^{4}-t^{2} T+4 T^{2}\right )+17 \ln \left (-\sqrt {t^{2}+T}\, t +2 T\right )-17 \ln \left (\sqrt {t^{2}+T}\, t +2 T\right )+\left (2 \,\operatorname {arctanh}\left (\frac {\left (t^{2}-8 T\right ) \sqrt {17}}{17 t^{2}}\right )+2 \,\operatorname {arctanh}\left (\frac {\left (t -4 \sqrt {t^{2}+T}\right ) \sqrt {17}}{17 t}\right )-2 \,\operatorname {arctanh}\left (\frac {\left (4 \sqrt {t^{2}+T}+t \right ) \sqrt {17}}{17 t}\right )\right ) \sqrt {17}-c_{1} = 0 \]

✓ Mathematica. Time used: 0.277 (sec). Leaf size: 135

ode=Sqrt[t^2+T[t]]==D[T[t],t]; 
ic={}; 
DSolve[{ode,ic},T[t],t,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\frac {1}{34} \left (-34 \log \left (\sqrt {t^2+T(t)}-t\right )-\left (\sqrt {17}-17\right ) \log \left (2 \left (\sqrt {17}-4\right ) t \sqrt {t^2+T(t)}-2 \left (\sqrt {17}-4\right ) t^2-\left (\sqrt {17}-3\right ) T(t)\right )+\left (17+\sqrt {17}\right ) \log \left (2 \left (4+\sqrt {17}\right ) t \sqrt {t^2+T(t)}-2 \left (4+\sqrt {17}\right ) t^2-\left (3+\sqrt {17}\right ) T(t)\right )\right )=c_1,T(t)\right ] \]

✗ Sympy

from sympy import * 
t = symbols("t") 
T = Function("T") 
ode = Eq(sqrt(t**2 + T(t)) - Derivative(T(t), t),0) 
ics = {} 
dsolve(ode,func=T(t),ics=ics)
 

NotImplementedError : The given ODE -sqrt(t**2 + T(t)) + Derivative(T(t), t) cannot be solved by the factorable group method